Definition with symbols

Relation with other notions

An equivalence relation is termed a congruence if it is both a left congruence and a right congruence.

Correspondence between subgroups and right congruences

The following is true:

Right congruences are precisely the equivalence relations whose equivalence classes are the right cosets of a subgroup

Proving that any right congruence gives right cosets

We first show that the equivalence class of the identity element is a subgroup. For this, we show the following three things:

Identity elements:The identity element is equivalent to the identity element: This follows on account of the relation being reflexive

Closure under multiplication: If , so is : The proof of this comes as follows. Suppose . Then . We already know that . Hence, by the transitivity of , we have .

Closure under inverses: If , then we can right multiply both sides by and obtain

Let denote this subgroup. Then clearly, for any , (right multiplying by ). Thus all the elements in the right coset of are in the same equivalence class as .

Further, we can show that if , they must be in the same right coset. Suppose . Then, right multiply both sides by . This gives , hence or .

Proving that right cosets give a right congruence

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